Quantifying Microbial Assembly Rules via Competitive Preference Embeddings

Microbial community assembly is governed by interaction rules whose complexity dictates stability and coexistence. Simple, transitive competitive hierarchies (e.g., A > B > C) imply a low-dimensional ecological niche space, whereas the existence of intransitivity (cyclic preferences) implies competitions that take place in a higher effective dimension. To identify the dimensionality of bacterial competition space, we use a generalization of the Bradley-Terry model from decision theory. Specifically, we construct a pairwise competition matrix between bacteria, either from relative abundance data or from in vitro experimental assays. We approximate these interaction matrices using lower dimensional embeddings and find that while bacteria can in principle compete over thousands of nutrients, effective nche dimensionality for bacterial competition is very small (~1-10). This dimensionality provides a direct, quantitative measure of the prevalence of intransitivity within the community's assembly rules. We further relate the competition dimensionality with ecosystem structure, including niche overlap, diversity, and environmental complexity. By integrating the results with established collective metrics (singular value spectra, $\beta/\alpha$ diversity relations). This work establishes a rigorous framework to distinguish between simple hierarchical models and more complex, multi-dimensional models of microbial competition.